Extended Kalman Filter Model for Gps and Indoor Positioning System

The most popular positioning system in the world is the Global Positioning System (GPS). However, GPS has a limited degree of accuracy for low priority users. This paper proposes a solution for solving these limitations by using the Extended Kalman Filter (EKF). Moreover, GPS is almost invalid in indoor environments. The paper also introduces an indoor positioning system model based on GPS ideology and EKF algorithm. Because of the similarity in ideology, it’s easier for the handoff procedure between outdoor and indoor environments and brings back the spatial continuous in positioning. The simulation results show that with the EKF, the accuracy of positioning is improved significantly in both outdoor and indoor environments. Keywords- GPS, Kalman filter, RFID, RSSI, EKF, indoor positioning.
I. INTRODUCTION
II. EKF MODEL FOR GPS

Generally, the Kalman algorithm is a group of mathematical equations described an efficient recurrence method for state estimation of the process that it is optimal in the sense that it minimizes the estimated error covariance when some presumed conditions are met [2]. EKF is an extension of the Kalman filter for non-linear systems.

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A. Global Positioning System (GPS). In order to positioning, it requested that user’s receivers get signals from at least 4 GPS satellites. Distances between users and satellites are determined by using the pseudo-range code. At the same time, satellites and receiver transmit the same pseudo-range code. Because of propagation delay, signals received from satellites have phases delay than a signal of receiver. By compared their phase, the distances can calculate. This method is called the Time of Arrival (TOA). [1]
B. Problem Expression. Positioning based on GPS is affected by many noise sources, such as propagated errors, satellite and receiver caused errors, other errors from Selective Availability, dilution of precision, interference, etc [1]. Several techniques are used to improve the accuracy of positioning in GPS, for example, DGPS (Differential GPS), Smart Antenna, Kalman Filter, etc. This paper focuses on the Extended Kalman Filter (EKF) solution, in order to introduce one way to model GPS system and sources of error. Nowadays, positioning applications in indoor environments are being extended. Especially, it becomes necessary in tunnels, supper huge plants, very high buildings, etc. The paper introduces a kind of Indoor Positioning System based on GPS’s ideology and using the EKF algorithm to help this system improve the accuracy of positioning. Assume that the GPS system is tracking a mobile object. It is a uniform speed motion in 3D space with attendance of random acceleration events. GPS receiver puts on object updates its position continuously. However, the location is affected by measurement noises and propagation noises. Therefore, the calculating position and the real position are different. In order to improve the accuracy of position, we use EKF to model system and noises so that it diminishes the effect of noises.
C. Modeling of system Defining the sate vector of system as follow:  RX ( n ) RY ( n )  RZ ( n )  X (n) =  VX ( n ) ? ?VY ( n ) VZ ( n ),? where RX(n),RY(n),RZ(n) are coordinates of user at nth sample, VX(n), VY(n), VZ(n)) are x, y, z – components of user’s velocity at nth sample. Following [3], the characteristic equations for the system can be extended as RX(n+1) = RX(n) + VX(n)T + ax(n)T 2 RY(n+1) = RY(n) + VY(n)T + ay(n)T 2 RZ(n+1) = RZ(n) + VZ(n)T + az(n)T 2 VX(n+1) = VX(n) + ax(n)T VY(n+1) = VY(n) + ay(n)T VZ(n+1) = VZ(n) + az(n)T (1) (2) (3) (4) where VX(n), VY(n), VZ(n)) are x, y, z – components of user’s velocity at nth sample, bf = ? bu/? t, dPRi is called delta-pseudorange correlated with user and ith satellite. [4] In order to reducing effect of errors, the EKF is used to model state noise vectors and measurement noise vectors. After characteristic matrixes are calculated, EKF iteration loops are started. The EKF algorithm will calculate estimation of the state vector by minimizing the estimated error covariance (between estimated values and real values).
D. Simulation results for outdoor-EKF GPS Data for simulation on Matlab7. 8. 0(R2009a).  User’s initial velocity: (3,6,2) meters/second Sampling rate: 1000 samples/second Iteration steps: 500 Process noise vector: W = 5* NORMRND (0, 500, 3, 1) Process noise variance: Q = 50* eye (3) Measurement noise vector: V = 5 * NORMRND (0, 500, 2, 1) Measurement noise variance: R = 50 * eye (2) (5) (6) here ax(n), ay(n), az(n) represent acceleration events at nth sample ( it is referred to state noises or process noises). According to [4], User’s positions are determined to base on the distances between the user and four satellites. PRi= +bu, i=1,2,3,4 (7). Fig. 1 shows the simulation results for Lagrange iteration and EKF iteration compared with the true position values. In geographic coordinates, PRI is the distance between the user and the ith satellite, (SXi, SYi, SZi) are coordinates of the ith satellite, (RX,RY,RZ) are coordinates of the user; bu=c. t with t is receiver clock offset compared to GPS time and c is the speed of light.

PRis is determined by GPS’s receiver. Coordinates of satellites are obtained by decoding satellite report, while (RX, RY, RZ) and bu are unknowns. With the system of equations (7) above, the root [RX, RY, RZ, bu,] can be calculated by using Lagrange iteration [4]. However, measurement values PRI are affected by noises (measurement noises). Therefore the root of the system of equations is not accuracy. After differentiating equations (7), we obtain dPRi = (RX ? SXi)? RX + (RY ? SYi)? RY + (RZ ? SZi)? RZ (RX ? SXi)2 + (RY ? SYi)2 + (RZ ? SZi)2 + ? bu = (RX ? SXi). VRX +(RY ? SYi). VRY +(RZ? SZi). VRZ +bf PRi.
Figure 1. Simulate tracking User’s trajectory in the outdoor environment (8) In Fig. 1, the red curve simulates the user’s motion, the Green curve simulates the calculated trajectory of the user receiver without EKF, the blue curve simulates calculated trajectory of the user receiver in the EKF model. Based on GPS’s ideology, this paper introduces an indoor positioning model using EKF, called Indoor-EKF GPS. Indoor-EKF GPS is hoped that it makes over easier with GPS, in such a way, we just use handling equipment to keep continuous positioning while moving between indoor-outdoor environments.
In Fig. 3, and an arranged system of equipment in space can be recognized as pseudo-satellites. Indoor spaces are complicated environments for wave propagation. Distances between user and pseudo-satellites cannot be determined using the TOA technique like in outdoor GPS. Here, the TOA technique is replaced by RSS (Received Signal Strength) technique. This technique measures the path loss and calculates the distance between the source and receiver.
Figure 2. Errors in outdoor positioning. Red points: positioning errors without EKF. Green points: positioning errors in the EKF model.
Comments on simulation results:
The maximum error is about 5 meters in case using the EKF model, whereas 25 meters in the case without EKF. Trajectory of a user receiver in the EKF model is closer to the trajectory of the user’s motion than trajectory of the user receiver without EKF. The average estimation error of EKF is very small than without the EKF case. However, several points in the curve are suddenly changing errors.
Figure 3. Indoor positioning system.
According to IEEE 802. 11 recommended channel model, the relation between free space path loss and distanced in breakpoint radius is given by [5] LFS(d) = L0 + 10? 1lg(d), 0 < d ? BP (9). According to the result, it shows that the positioning errors are reduced significantly. III. A. INDOOR POSITIONING SYSTEM where ? 1 is called distance-power-gradient up to breakpoint distance DBP, Lo represents the path loss in decibels at a one-meter distance. The overall path loss for any distance is modeled as [5] Indoor-EKF GPS Recent years, the indoor environment has been extended so that indoor positioning demands are extended, too. Furthermore, it becomes necessary in tunnels, supper huge plants, very high buildings, etc, and giving convenient for absent minder when household equipment are positioning.
However, GPS is almost invalid in an indoor environment. The reason is that the GPS signal has low power. Even the GPS signal can be received, the error positioning of GPS is not appropriate with Indoor applications. ?LFS (d ) +W, d? DBP? L(d ) = ? ? d? ?LFS (DBP ) +10? 2 LG ? d? +W, d > DBP ? BP ? ? (10) where ? 2 is distance-power-gradient over break-point distance DBP. It’s required at least four distances form user to pseudo satellites are determined for calculating the user’s coordinates. Fig. 4 shows the calculation process for the user position.
Figure 6. The second model.
Figure 4. The user coordinates calculation.
Reference

[1]. Ahmed EI-Rabbani, “Introduction to GPS”, Artech House, Inc, ISBN 1-58053-183-0, 2002, pp. 13-25,2741.
[2]. Grey Welch and Gary Bishop, “An introduction to the Kalman filter”, Technical Report TR 95-041, 2001.
[3]. Jorge Quijano, “Estimation of the position of a moving target using Extended Kalman Filter”, term paper for the class ECE 510 Statistical Signal Processing, winter 2006.
[4]. James Bao-Yen Tsui, “Fundamentals of Global Positioning System Receivers: A Software Approach”, John Wiley & Sons, Inc, ISBN 0-471-20054-9, 2000, pp. 9-15, 230-231.
[5].  Ahmad Hatami, “Application of Channel Modeling for Indoor Location Using TOA and RSS”, Ph.D. Thesis, Worcester Polytechnic Institute, 2006, pp. 14-19.

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